Square One: In Phase

If you've been shopping for an equalizer (EQ) recently, you may have noticed that many products, especially those designed for mastering use, describe themselves as linear-phase EQs. But what does phase have to do with EQ, and what exactly is linear about linear-phase designs? In this article, I'll answer those questions and explain why such designs are touted for their transparency.

A Pretty Phase

Equalizers are composed of filters, and filters are components that alter the frequency content of a signal. (For more on EQs, see “Square One: The Sculptor's Tool” in the December 2007 issue, available at emusician.com.) It's a basic function of analog filter design that frequencies within the range passing through a filter are delayed by differing amounts. Time delay in a signal means phase shift, and changes in phase can be audible.

Exactly when phase changes are audible is a bit thornier to pin down. Our ears are very sensitive to modulated changes in phase, such as in chorus, flange, and other effects. We can clearly hear two copies of identical frequencies canceling each other, as when one speaker is wired backward. We can also hear the in-and-out-of-phaseness of two closely related frequencies, such as the beat frequency heard when tuning an instrument (or failing to do so). However, it's much more difficult to hear the difference between two complex signals that have the same spectral content but different phase relationships between their components.

Nevertheless, there are times when this phase shifting becomes audible, perhaps most notably on transients, the short, attack portions of a sound. For these situations, it is desirable to build filters that change the complex phase relationships of a sound's components as little as possible.

Analog filters are generally designed to be minimum-phase devices, meaning that they introduce as little total phase shift as possible. The amount of phase shift varies according to the filter design, and typically more aggressive filters (those with steeper slope) introduce more phase shift. This is why linear-phase EQs were born.

Phasing Facts

FIG. 1: Phase delay in a linear-phase filter is proportional to the frequency, so as the frequency gets higher, the phase delay (in degrees) gets longer. Because the period gets shorter, more degrees of phase shift end up taking the same amount of time.
Chuck Dahmer

Linear-phase filters still introduce phase shift in a signal, but the phase shift is proportional to the frequency. In other words, the phase changes more as the frequency increases. Because a waveform's period (the length of time for one iteration, or cycle) decreases as frequency increases, the increase in phase shift balances the decrease in period, resulting in a constant delay across all frequencies (see Fig. 1). Because all frequencies are delayed by the same amount, the essential waveform of the signal is kept intact. This preserves transients, which, as mentioned, are more vulnerable to the effects of phase shift. Whatever phase issues our ears might pick up are mitigated by linear-phase filters.

Linear-phase response is especially valuable where multiple filters are used to split a signal, as in multiband compressors and de-essers. With minimum-phase designs, each band subjects the signal to phase shift at each point where bands overlap (twice, actually — once by the upper band and once by the lower). Even when no gain is applied to a band, the signal is nevertheless changed. With linear-phase filters, the device can be truly transparent when no gain is applied. When gain is applied to a band, the effect is limited to the desired increase or decrease of that band.

There are, of course, times when transparency is not desirable, and for that there are plenty of EQ designs with ample character, not all of which comes from phase shift. The DSP required for linear-phase EQs also subjects the signal to a significant delay, which makes them poor choices for track inserts or live sound-reinforcement applications. On the mix bus, this additional latency is usually inconsequential.

Nailing It

As with carpentry, audio engineering is about choosing the right tool for the job. No self-respecting carpenter would go out on a job with only a ball-peen hammer, and no engineer would declare a single type of EQ to be universally superior to other types. Linear-phase EQs preserve the characteristic shape of a waveform by shifting all frequencies by the same amount, making them ideal for applications such as mastering, where subtlety and transparency are paramount and latency is acceptable.